A man can row 8 km/hr in still water. When the river is running at 2 km/hr, it takes him 3 hrs 12 mins to row to a place and back. How far is the place?

Options

A. 12 km
B. 15 km
C. 18 km
D. 10 km

Answer

To solve the problem step by step, we will break down the information given and apply the relevant formulas.

### Step 1: Understand the problem
The man can row at a speed of 8 km/hr in still water. The river flows at a speed of 2 km/hr. We need to find out how far the place is, given that it takes him 3 hours and 12 minutes to row to the place and back.

### Step 2: Convert time into hours
First, we convert the total time from hours and minutes into hours only.
3 hours and 12 minutes can be converted as follows:
- 12 minutes = 12/60 hours = 1/5 hours
- Therefore, total time = 3 + 1/5 = 16/5 hours.

### Step 3: Calculate effective speeds
When rowing downstream (with the current), the effective speed is:
- Downstream speed = Speed in still water + Speed of the river = 8 km/hr + 2 km/hr = 10 km/hr.

When rowing upstream (against the current), the effective speed is:
- Upstream speed = Speed in still water - Speed of the river = 8 km/hr - 2 km/hr = 6 km/hr.

### Step 4: Set up the equation
Let the distance to the place be \(d\) km. The time taken to row downstream to the place is:
- Time downstream = Distance / Speed = \(d / 10\) hours.

The time taken to row upstream back is:
- Time upstream = Distance / Speed = \(d / 6\) hours.

The total time for the round trip is:
\[ \frac{d}{10} + \frac{d}{6} = \frac{16}{5} \text{ hours}. \]

### Step 5: Find a common denominator and solve for \(d\)
The least common multiple (LCM) of 10 and 6 is 30. We can rewrite the equation as:
\[ \frac{3d}{30} + \frac{5d}{30} = \frac{16}{5}. \]
Combining the fractions gives:
\[ \frac{8d}{30} = \frac{16}{5}. \]

### Step 6: Cross-multiply to solve for \(d\)
Cross-multiplying gives:
\[ 8d \cdot 5 = 16 \cdot 30. \]
This simplifies to:
\[ 40d = 480. \]
Dividing both sides by 40 gives:
\[ d = \frac{480}{40} = 12 \text{ km}. \]

### Conclusion
The distance to the place is **12 kilometers**.